A conjecture for q-decomposition matrices of cyclotomic v-Schur algebras

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A conjecture for q-decomposition matrices of cyclotomic v-Schur algebras

Xavier Yvonne

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Xavier Yvonne. A conjecture for q-decomposition matrices of cyclotomic v-Schur algebras. 2006.

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A conjecture for q-decomposition matrices of cyclotomic v-Schur algebras

Xavier YVONNE April 27, 2006

Abstract

The Jantzen sum formula for cyclotomic v-Schur algebras yields an identity for some q-analogues of the decomposition matrices of these algebras. We prove a similar identity for matrices of canonical bases of higher-level Fock spaces. We conjecture then that those matrices are actually identical for a suitable choice of parameters. In particular, we conjecture that decomposition matrices of cyclotomic v-Schur algebras are obtained by specializing at q = 1 some transition matrices between the standard basis and the canonical basis of a Fock space.

1 Introduction

In order to study representations of the Ariki-Koike algebra associated to the complex reflec- tion group G(l, 1, m), Dipper, James and Mathas introduced in 1998 the cyclotomic v-Schur algebra [DJM]. This algebra depends on the two integers l and m and on some deformation parameters v, u ₁ , . . . , u _l . When l = 1, the cyclotomic v-Schur algebra coincides with the v-Schur algebra of [DJ]. It is an open problem to calculate the decomposition matrix of a cyclotomic v-Schur algebra whose parameters are powers of a given n-th root of unity. To this aim, James and Mathas proved, for cyclotomic v-Schur algebras, an important formula:

the Jantzen sum formula [JM]. Given a Jantzen filtration for Weyl modules, one can define a q-analogue D(q) of the decomposition matrix; the coefficients of D(q) are graded decompo- sition numbers of the composition factors of Weyl modules (see Definition 2.5). The Jantzen sum formula is equivalent to the identity D ^′ (1) = J ^⊳ D(1), where J ^⊳ is a matrix of ℘-adic valuations of factors of some Gram determinants (see Theorem 2.3 and Corollary 2.7).

Let ∆(q) be the matrix of the canonical basis of the degree m homogeneous component of a Fock representation of level l of U q ( sl b _n ) [U2]. Uglov provided in [U2] an algorithm for computing ∆(q).

In view of Ariki’s theorem for Ariki-Koike algebras [A2], it seems natural to conjecture

that for a suitable choice of parameters, one has D(q) = ∆(q). This would provide an algo-

rithm for computing decomposition matrices of cyclotomic v-Schur algebras. Varagnolo and

Vasserot [VV] proved for l = 1 that D(1) = ∆(1). Moreover, Ryom-Hansen showed that

this conjecture (still for l = 1) is compatible with the Jantzen-Schaper formula [Ry]. Passing

to higher level l ≥ 1 requires the introduction of an extra parameter s l = (s ₁ , . . . , s l ) ∈ Z ^l ,

called multi-charge; this l-tuple parametrizes the Fock space of level l introduced by Uglov.

We say that s l is m-dominant if for all 1 ≤ d ≤ l − 1, we have s _d+1 − s d ≥ m. In this case, we conjecture that D(q) = ∆(q). Here, D(q) comes from a Jantzen filtration of the Weyl modules of the cyclotomic v-Schur algebra S C = S C ,m (ζ ; ζ ^s

, . . . , ζ ^s

) with ζ := exp( ^2iπ _n ).

Note that for any choice of roots of unity ζ ^r

, . . . , ζ ^r

(that is, for any r ₁ , . . . , r l ∈ Z /n Z ) and any m we can find an m-dominant multi-charge s l = (s ₁ , . . . , s l ) such that ζ ^s

= ζ ^r

(1 ≤ d ≤ l). Therefore, putting q = 1, our conjecture gives an algorithm for calculating the decomposition matrix of an arbitrary cyclotomic v-Schur algebra S C = S C ,m (ζ ; ζ ^s

, . . . , ζ ^s

).

Such a conjecture is new even for type B m (case l = 2).

Our conjecture is supported by the following theorem. We define in a combinatorial way a matrix J ^≺ for any multi-charge s l ; if s l is m-dominant, then our matrix J ^≺ coincides with the matrix J ^⊳ of the Jantzen sum formula. We show then that for any multi-charge s _l , we have ∆ ^′ (1) = J ^≺ ∆(1) (Theorem 2.8).

The proof of our theorem relies on a combinatorial expression for the derivative at q = 1 of the matrix A(q), where A(q) is the matrix of the Fock space involution used for defining

∆(q). Namely, we show that A ^′ (1) = 2J ^≺ (Theorem 2.11). The coefficients of A(q) are some analogues for Fock spaces of Kazhdan-Lusztig R-polynomials R x,y (q) for Hecke algebras. The classical computation of R _x,y ^′ (1) was made in [GJ], in relation with the Kazhdan-Lusztig con- jecture for multiplicities of composition factors of Verma modules.

Acknowledgments. I would like to thank Nicolas Jacon and my advisor Bernard Leclerc for inspiring discussions about Ariki-Koike algebras. I also would like to thank Bernard Leclerc for his assistance and constant advice when I was writing this article. At last, I thank Andrew Mathas, Hyohe Miyachi and the referee for their comments.

Notation 1.1 Let N (resp. N ^∗ ) denote the set of nonnegative (resp. positive) integers, and for a, b ∈ R denote by [[a; b]] the discrete interval [a, b] ∩ Z . Throughout this article, we fix three integers n, l, m ≥ 1. Let Π be the set of partitions of any integer and Π ^l _m be the set of l-multi-partitions of m. The Coxeter group of type A _r−1 (with r ∈ N ^∗ ) is the symmetric group S _r = hσ i = (i, i + 1) | 1 ≤ i ≤ r − 1i. Let ℓ be the length function on S _r and ω be the

unique element of maximal length in S _r . ⋄

PART A: Statement of results

2 Statement of results

2.1 The Jantzen sum formula

Definition 2.1 ([AK, BM]) Let R be a principal ideal domain. Let v be an invertible element of R and u ₁ , . . . , u _l ∈ R. The Ariki-Koike algebra, denoted by

(1) H = H R = H R,m (v; u ₁ , . . . , u _l ),

is the algebra defined over R with generators T 0 , . . . , T m−1 and relations

(2)

 

 

 

 

 



(T ₀ − u ₁ ) · · · (T ₀ − u _l ) = 0,

T ₀ T ₁ T ₀ T ₁ = T ₁ T ₀ T ₁ T ₀ ,

(T i + 1)(T i − v) = 0 (1 ≤ i ≤ m − 1), T _i T _i+1 T _i = T _i+1 T _i T _i+1 (1 ≤ i ≤ m − 2),

T i T j = T j T i (0 ≤ i < j − 1 ≤ m − 2).

⋄ Following [DJM], let

(3) S = S R = S R,m (v; u ₁ , . . . , u l )

be the cyclotomic v-Schur algebra associated to H. Dipper, James and Mathas (see [DJM, Theorem 6.12]) showed that S is a cellular algebra in the sense of [GL]. Given λ l ∈ Π ^l _m , one defines as in [DJM, Definition 6.13] a right S-module W (λ _l ) which is a free R-module of finite rank, called Weyl module. Since S is cellular, W (λ l ) is naturally equipped with a symmetric bilinear form h·, ·i. Set

(4) L(λ l ) := W (λ l )/rad W (λ l ),

where rad W (λ l ) is the radical of the bilinear form h·, ·i. Assume temporarily that R is a field. By [DJM, Corollary 6.18], S is a quasi-hereditary algebra, so the theory of cellular al- gebras of [GL] shows that {L(λ l ) | λ _l ∈ Π ^l _m } is a complete set of non-isomorphic irreducible S-modules (see [DJM, Theorem 6.16]). This implies that R 0 (S), the Grothendieck group of finitely-generated S-modules, is a free Z -module with basis {[L(λ l )] | λ l ∈ Π ^l _m }.

From now on, we assume that R is a local ring, with unique maximal ideal ℘. Let ν ℘

be the corresponding ℘-adic valuation map. Let K be the field of fractions of R and extend ν ℘ to K in the natural way. Let F = R/℘R be the residue field, so (R, K, F ) is a modular system. If M is a right R-module, we denote by M F = M ⊗ R F the specialized module and denote similarly by M K = M ⊗ R K the corresponding module defined over K. We shall use this notation for Weyl modules and for S itself.

Definition 2.2 ([Jan], see also [AM]) Let M be an R-module equipped with a symmetric bilinear form h·, ·i. For all i ∈ N , set

(5) M (i) := {u ∈ M | ∀ v ∈ M, ν ℘ (hu, vi) ≥ i}.

The Jantzen filtration of M is the sequence

(6) M _F = M _F (0) ⊃ M _F (1) ⊃ · · · ,

where M F (i) := (M(i) + ℘M )/℘M . ⋄

Note that in the definition above, we have in particular M F (1) = rad M F . Moreover, if M is free of finite rank (as an R-module), then we have M F (i) = {0} for i large enough.

The following theorem was proved by James and Mathas (see [JM, Theorem 4.3]).

Theorem 2.3 (the Jantzen sum formula)

Assume that S K is semisimple. Then in the Grothendieck group R 0 (S F ), we have for all λ _l ∈ Π ^l _m :

(7) X

i>0

[W F (λ l ; i)] = X

µ

∈Π

ν ℘ (g λ

,µ

) [W F (µ _l )].

Here, the g _λ

_,µ

∈ R are factors of some Gram determinants (see [JM, Definitions 3.1, 3.36

and Corollary 3.38]).

Remark 2.4 The condition of semisimplicity of S K is stated in [JM, Theorem 4.3] in terms of the Poincar´e polynomial for H R , which is defined in [JM, Definition 3.40]. ⋄ James and Mathas [JM] showed that only multi-partitions µ _l ∈ Π ^l _m such that µ _l ⊳ λ _l contribute to the right hand-side of Theorem 2.3 (the definition of the dominance ordering

⊳ is recalled in Definition 3.1). They have given a combinatorial expression of ν ℘ (g _λ

_,µ

) in terms of ribbons contained in diagrams of l-multi-partitions. However, this combinatorial expression makes sense even if λ l does not dominate µ _l . We will therefore introduce in Section 3.4 a matrix J = j _λ

_,µ

λ

,µ

∈Π

whose entries are these combinatorial expressions without restriction on the pair (λ _l , µ _l ). More precisely, our indexing is chosen so that

(8) j _λ

l

,µ

= ν _℘ (g _λ

_,µ

) if µ _l ⊳ λ _l ,

where the sign † denotes the conjugation of multi-partitions (see (19)). We are forced to use conjugates here because the indexation from [JM] for the rows and columns of decomposi- tion matrices is not compatible with the indexation from [U2] for the rows and columns of transition matrices for Uglov’s canonical bases.

Now, let 6 be an arbitrary partial ordering on Π ^l _m and write λ l < µ _l if λ l 6 µ _l and λ _l 6= µ _l (λ _l , µ _l ∈ Π ^l _m ). Define a matrix J ^< = j _λ ^<

l

,µ

λ

,µ

∈Π

by the formula

(9) j _λ ^<

l

,µ

:=

( j _λ

,µ

if λ l < µ _l

0 otherwise (λ _l , µ _l ∈ Π ^l _m ).

If we take 6 = E , then we get a matrix J ^⊳ whose entries are, up to conjugation of multi- partitions, the ν ℘ (g _λ

,µ

)’s of [JM].

We now derive a matrix identity equivalent to the Jantzen sum formula.

Definition 2.5 Let D(q) = d λ

,µ

(q)

λ

,µ

∈Π

be the matrix defined by (10) d λ

,µ

(q) := X

i≥0

W F (λ ^† _l ; i)/W F (λ ^† _l ; i + 1) : L F (µ ^† _l )

q ⁱ ∈ N [q] (λ l , µ _l ∈ Π ^l _m ).

⋄

Note that d _λ

l

,µ

(1) is equal to the multiplicity of L F (µ _l ) as a composition factor of W F (λ l ), so up to conjugation of multi-partitions (which amounts to reindexing the rows and columns of the matrix), D(1) is the usual decomposition matrix of S F .

Lemma 2.6 Let M = m _λ

,µ

λ

,µ

∈Π

be a matrix with integer entries. Then the following statements are equivalent:

(i) In R 0 (S F ), we have for all λ _l ∈ Π ^l _m : X

i>0

[W F (λ ^† _l ; i)] = X

ν

∈Π

m _λ

_,ν

[W F (ν ^† _l )], (ii) D ^′ (1) = M D(1).

Proof. Let λ l ∈ Π ^l _m . Since {

L F (µ ^† _l ) µ _l ∈ Π ^l _m } is a Z -basis of R 0 (S F ), we have on the one hand:

X

i>0

W _F (λ ^† _l ; i)

= X

i>0

X

µ

∈Π

W _F (λ ^† _l ; i) : L _F (µ ^† _l )

L _F (µ ^† _l )

= X

µ

∈Π

X

i>0

X

j≥i

W F (λ ^† _l ; j) / W F (λ ^† _l ; j + 1) : L F (µ ^† _l )

L F (µ ^† _l )

= X

µ

∈Π

X

j>0

X

0<i≤j

W F (λ ^† _l ; j) / W F (λ ^† _l ; j + 1) : L F (µ ^† _l )

L F (µ ^† _l )

= X

µ

∈Π

d ^′ _λ

l

,µ

(1)

L _F (µ ^† _l ) .

On the other hand, we have X

ν

∈Π

m _λ

,ν

W F (ν ^† _l )

= X

µ

, ν

∈Π

m _λ

,ν

W F (ν ^† _l ) : L F (µ ^† _l )

L F (µ ^† _l )

= X

µ

∈Π

X

ν

∈Π

m _λ

_,ν

W F (ν ^† _l ) : L F (µ ^† _l )

L F (µ ^† _l )

= X

µ

∈Π

X

ν

∈Π

m _λ

_,ν

d _ν

_,µ

(1)

L _F (µ ^† _l ) .

since the

L _F (µ ^† _l )

, µ _l ∈ Π ^l _m are linearly independent, the lemma follows.

The Jantzen sum formula as stated in Theorem 2.3, together with (8) and Lemma 2.6, implies the following result.

Corollary 2.7 Assume that S K is semisimple. Then with the notation above, we have

(11) D ^′ (1) = J ^⊳ D(1).

2.2 Statement of theorems

In this section, we state an important conjecture for computing the decomposition matrix of the cyclotomic v-Schur algebra defined over C , with parameters equal to arbitrary powers of a primitive n-th root of unity. This conjecture is supported by Theorem 2.8.

2.2.1 Choice of parameters

Fix (r ₁ , . . . , r l ) ∈ ( Z /n Z ) ^l . We shall define a modular system (R, K, F ) with parameters such that the specialized cyclotomic v-Schur algebra S F is S ^C ,m (ζ; ζ ^r

, . . . , ζ ^r

) with ζ := exp( ^2iπ _n ).

We first define a modular system (R, K, F ) as follows. Let R b = C [x, x ⁻¹ ] be the ring of Laurent polynomials in one indeterminate over the field C . Let

(12) ξ := exp

2iπ nl

∈ C , ℘ := (x − ξ), R := C [x, x ⁻¹ ] ℘ , K := C (x) and F := R/℘R ≃ C ,

that is, ℘ is the prime ideal in R b spanned by x − ξ with ξ a primitive complex nl-th root of unity, R is the localized ring of R b at ℘, K is the field of fractions of R and F is the residue field.

Following [U2], we fix an l-tuple s _l in

(13) L(r 1 , . . . , r l ) := {(s 1 , . . . , s l ) ∈ Z ^l | ∀ 1 ≤ d ≤ l, r d = s d mod n}.

Such an l-tuple is called a multi-charge. The multi-charge s l parametrizes a so-called (q- deformed) Fock space of level l, denoted by F q [s l ] (see Section 4.1). Note that for a given (r ₁ , . . . , r l ) ∈ ( Z /n Z ) ^l we have an infinite choice of Fock spaces F _q [s _l ] such that s _l is in L(r ₁ , . . . , r l ).

We now describe the choice of parameters for the cyclotomic v-Schur algebra S. These parameters are similar to those used in [Jac] for Ariki-Koike algebras. They depend on n, l and on the multi-charge (s ₁ , . . . , s _l ) ∈ L(r 1 , . . . , r _l ) that we have fixed. Put

(14) v := x ^l and u d := ξ ^nd x ^ls

^−nd (1 ≤ d ≤ l).

Note that we have S F = S C ,m (ζ; ζ ^r

, . . . , ζ ^r

) with ζ := exp( ^2iπ _n ). Note also that the algebra S K,m (v; u ₁ , . . . , u l ) is semisimple. Indeed, specializing x at 1 sends H K,m (v; u ₁ , . . . , u l ) on the semisimple group algebra C G(l, 1, m), so by the Tits deformation argument [A1], the algebra H K,m (v; u 1 , . . . , u l ) is semisimple and so is S K,m (v; u 1 , . . . , u l ). Therefore, the Jantzen sum formula (see Theorem 2.3) applies in our case. This leads in particular to the definition of a matrix J ^⊳ (see Section 3.4).

2.2.2 Main result

Following [U2], let s _l ∈ L(r 1 , . . . , r _l ) and F q [s _l ] be the corresponding Fock space of level l (see

Section 4.1). As a vector space, F _q [s _l ] has a natural basis {|λ l , s _l i | λ _l ∈ Π ^l } and a canonical

basis {G(λ l , s l ) | λ _l ∈ Π ^l } indexed by l-multi-partitions. Let F q [s l ] m be the subspace of F q [s l ] spanned by the |λ l , s l i’s, λ l ∈ Π ^l _m . Let A(q) be the matrix of the involution of F q [s l ] m with respect to the standard basis, and let ∆(q) be the transition matrix between the standard basis and the canonical basis of F q [s l ] m (see Sections 4.2 and 4.3). Still following [U2], we associate to s l an ordering ≺ (see Definition 3.10). By (9) we get a matrix J ^≺ .

Theorem 2.8 Let s _l ∈ L(r ₁ , . . . , r l ). Then with the notation above, we have

(15) ∆ ^′ (1) = J ^≺ ∆(1).

Example 2.9 Take n = 3, l = 2, s _l = (1, 0) and m = 3. Then we have on the one hand

J ^≺ =



 

 

 

 

 

 

 



0 . . . . . . . . .

0 0 . . . . . . . .

0 0 0 . . . . . . .

0 0 0 0 . . . . . .

0 0 1 0 0 . . . . .

0 1 1 0 0 0 . . . .

0 −1 0 0 1 1 0 . . .

0 1 −1 0 1 −1 1 0 . .

0 1 −1 0 0 1 0 0 0 .

0 −1 0 0 −1 0 1 0 1 0



 

 

 

 

 

 

 



(1, 1), (1) (3), ∅

∅, (3) (1), (2)

∅, (2, 1) (2), (1) (1), (1, 1)

∅, (1, 1, 1) (2, 1), ∅ (1, 1, 1), ∅

,

where dots over the main diagonal stand for zero entries. The l-multi-partitions of m which index the bases of F _q [s _l ] _m are ordered decreasingly with respect to a total ordering finer than

≺ and they are displayed in the column located on the right of the matrix J ^≺ . On the other hand, we compute ∆(q) using Uglov’s algorithm (see [U2]). If we keep the same ordering for the rows and the columns of ∆(q), we get the following matrix.

∆(q) =



 

 

 

 

 

 

 



1 . . . . . . . . .

0 1 . . . . . . . .

0 0 1 . . . . . . .

0 0 0 1 . . . . . .

0 0 q 0 1 . . . . .

0 q q 0 0 1 . . . .

0 0 q ² 0 q q 1 . . .

0 0 0 0 q ² 0 q 1 . .

0 q ² 0 0 0 q 0 0 1 .

0 0 0 0 0 q ² q 0 q 1



 

 

 

 

 

 

 



(1, 1), (1) (3), ∅

∅, (3) (1), (2)

∅, (2, 1) (2), (1) (1), (1, 1)

∅, (1, 1, 1) (2, 1), ∅ (1, 1, 1), ∅

.

It is easy to check that ∆ ^′ (1) = J ^≺ ∆(1). ⋄

Example 2.10 Take n = 3, l = 2, s l = (4, −3) and m = 3. Write the rows and the columns of the following matrices with respect to a total ordering finer than ≺. Then

J ^≺ =



 

 

 

 

 

 

 



0 . . . . . . . . .

0 0 . . . . . . . .

0 1 0 . . . . . . .

0 1 1 0 . . . . . .

0 −1 1 0 0 . . . . .

0 0 0 0 0 0 . . . .

0 −1 0 1 1 0 0 . . .

0 0 −1 1 0 0 0 0 . .

0 0 0 0 −1 0 1 1 0 . 0 1 0 −1 0 0 1 −1 1 0



 

 

 

 

 

 

 



(1, 1), (1) (3), ∅ (2, 1), ∅ (2), (1) (1, 1, 1), ∅ (1), (2) (1), (1, 1)

∅, (3)

∅, (2, 1)

∅, (1, 1, 1) and

∆(q) =



 

 

 

 

 

 

 



1 . . . . . . . . .

0 1 . . . . . . . .

0 q 1 . . . . . . .

0 q ² q 1 . . . . . .

0 0 q 0 1 . . . . .

0 0 0 0 0 1 . . . .

0 0 q ² q q 0 1 . . .

0 0 0 q 0 0 0 1 . .

0 0 0 q ² 0 0 q q 1 .

0 0 0 0 q 0 q ² 0 q 1



 

 

 

 

 

 

 



(1, 1), (1) (3), ∅ (2, 1), ∅ (2), (1) (1, 1, 1), ∅ (1), (2) (1), (1, 1)

∅, (3)

∅, (2, 1)

∅, (1, 1, 1) .

Again, one can check that ∆ ^′ (1) = J ^≺ ∆(1). ⋄

Theorem 2.8 is equivalent to the following:

Theorem 2.11 With the notation of Theorem 2.8, we have

(16) A ^′ (1) = 2J ^≺ .

Proof of the equivalence of Theorems 2.8 and 2.11. Since the canonical basis is invariant under the involution, we have ∆(q) = A(q)∆(q ⁻¹ ). Taking derivatives at q = 1 yields

∆ ^′ (1) = A ^′ (1)∆(1)−A(1)∆ ^′ (1). Since A(1) is the identity matrix, we get 2∆ ^′ (1) = A ^′ (1)∆(1).

As a consequence, Theorem 2.11 implies Theorem 2.8. Since ∆(1) is unitriangular, hence

invertible, the converse follows.

We prove Theorem 2.11 in Part C. Our proof is similar to the proof of [Ry] in the level one case. However the higher-level case is significantly more complicated and involves the discussion of many cases (see Section 7).

2.2.3 A conjecture for the decomposition matrix of S

Choose the parameters as in Section 2.2.1. Guided by the formal analogy between Theorem 2.8 on one hand, and the rephrasing of the Jantzen sum formula given in Corollary 2.7 on the other hand, we may wonder if for some s l ∈ L(r ₁ , . . . , r l ), the corresponding matrix J ^≺ coincides with the matrix J ^⊳ coming from the Jantzen sum formula. This leads to the following definition and conjecture.

Definition 2.12 Let M ∈ N . We say that s l ∈ L(r 1 , . . . , r l ) is M-dominant if for all 1 ≤ d ≤ l − 1, we have

(17) s _d+1 − s d ≥ M.

⋄ The point is that if s l is m-dominant, then we have J ^≺ = J ^⊳ (see Proposition 5.12).

Conjecture 2.13 Assume that s _l ∈ L(r 1 , . . . , r _l ) is m-dominant. Let D(q) be the q-analogue of the decomposition matrix of S defined in Definition 2.5 with our choice of parameters given in Section 2.2.1. Then we have

(18) D(q) = ∆(q).

If we put q = 1 in Conjecture 2.13, we thus get an algorithm for computing the decomposition matrix of S ^C ,m (ζ ; ζ ^r

, . . . , ζ ^r

) with ζ := exp( ^2iπ _n ).

Remark 2.14 The assumption of m-dominance is necessary in Conjecture 2.13. Indeed, while the decomposition matrix D(1) only depends on the sequence (r 1 , . . . , r l ) of the residues modulo n of the multi-charge s l , the matrix ∆(1) actually depends on s l itself. For example, take n = 3, l = 2 and m = 3. Then the multi-charges (1, 0) and (4, −3) are both in L(1, 0), but the corresponding matrices ∆(1) do not have the same number of zero entries (see Ex-

amples 2.9 and 2.10). ⋄

Remark 2.15 Conjecture 2.13 suggests that the matrix ∆(1) should not depend of the choice of the multi-charge s l ∈ L(r ₁ , . . . , r l ) provided it is M -dominant for M large enough.

This statement is proved in [Y, Th´eor`eme 4.30]), where an explicit value of M is given.

However, the fact that we might take M = m here is still conjectural. ⋄

Example 2.16 Set n = 3, l = 2, (r 1 , r 2 ) = (1, 0) and m = 3. Then the specialized cy- clotomic v-Schur algebra is S ^C ,3

e

; e

, 1

. Take s l = (4, −3), so s l ∈ L(r 1 , . . . , r l ) is m-dominant. According to Conjecture 2.13, we expect D(q) be equal to

∆(q) =



 

 

 

 

 

 

 



1 . . . . . . . . .

0 1 . . . . . . . .

0 q 1 . . . . . . .

0 q ² q 1 . . . . . .

0 0 q 0 1 . . . . .

0 0 0 0 0 1 . . . .

0 0 q ² q q 0 1 . . .

0 0 0 q 0 0 0 1 . .

0 0 0 q ² 0 0 q q 1 .

0 0 0 0 q 0 q ² 0 q 1



 

 

 

 

 

 

 



(1, 1), (1) (3), ∅ (2, 1), ∅ (2), (1) (1, 1, 1), ∅ (1), (2) (1), (1, 1)

∅, (3)

∅, (2, 1)

∅, (1, 1, 1)

(see Example 2.10). ⋄

If we no longer assume that s _l is m-dominant, then we expect ∆(q) be equal to a q- analogue of the decomposition matrix of a quasi-hereditary covering (in the sense of Rouquier, see [Ro]) of the Ariki-Koike algebra H. This covering, depending on s _l , could come from a rational Cherednik algebra through the Knizhnik-Zamolodchikov functor [GGOR]. It should be Morita-equivalent to the cyclotomic v-Schur algebra of [DJM] if s l is m-dominant.

PART B: Tools for the proof of Theorem 2.11

The next two sections recall some results about combinatorics of partitions and multi- partitions on the one hand and higher-level Fock spaces on the other hand; all of them will be used in the proof of Theorem 2.11. However, there are no new results here, so the reader familiar with these two topics may skip this part and come back to it later in order to get the needed definitions and notation.

3 Combinatorics of partitions and multi-partitions

3.1 Definitions

We give here all the basic definitions about partitions and multi-partitions that we need

later; our main reference is [Mac]. Let r ∈ N . A partition of r is a sequence of integers

λ = (λ ₁ , λ ₂ , . . . , λ N ) such that λ ₁ ≥ λ ₂ ≥ . . . ≥ λ N ≥ 0 and λ ₁ + · · · + λ N = r. Each nonzero

λ i is called a part of λ. The sum of all the parts of λ is denoted by |λ|. We identify two

partitions differing only by a tail of zeroes and write sometimes partitions as sequences of

integers with an infinite tail of zeroes. The only partition of 0 is denoted by ∅. The conjugate of the partition λ is the partition λ ^† defined by

(19) λ ^† _i := ♯{j | λ j ≥ i} (i ≥ 1);

for example, the conjugate of (4, 3, 3, 2, 1) is (5, 4, 3, 1).

An N -multi-partition of r is an N -tuple of partitions of integers summing up to r. Let λ = (λ ⁽¹⁾ , . . . , λ ^{(N )} ) be an N -multi-partition. The conjugate of λ is the multi-partition λ ^† := (λ ^(N) ) ^† , . . . , (λ ⁽¹⁾ ) ^†

. For 1 ≤ b ≤ N , write λ ^(b) = (λ ^(b) ₁ , λ ^(b) ₂ , . . .) the parts of λ ^(b) . The Young diagram of λ is the set

(20) {(i, j, b) ∈ N ^∗ × N ^∗ × [[1; N ]] | 1 ≤ j ≤ λ ^(b) _i },

whose elements are called boxes or nodes of λ. If N = 1, namely, if λ is a partition, we drop the third component in the symbol (i, j, b) of a node of λ. From now on we identify an N -multi-partition with its Young diagram. We extend the notation |λ| in a natural way for multi-partitions and define the dominance ordering on multi-partitions as follows.

Definition 3.1 Let λ and µ be two N -multi-partitions. We say that µ dominates λ and write λ E µ if

(21) |λ| = |µ|

and for all k ≥ 0, 1 ≤ b ≤ N , we have (22)

X b−1

i=1

|λ ⁽ⁱ⁾ | + X k

j=1

λ ^(b) _j ≤ X b−1

i=1

|µ ⁽ⁱ⁾ | + X k

j=1

µ ^(b) _j .

Write λ ⊳ µ if λ E µ and λ 6= µ. ⋄

If λ, µ ∈ Π are two partitions, write λ ⊂ µ if the diagram of λ is contained in the diagram of µ, and the set-theoretic difference is called a skew diagram; we denote it by µ/λ. A path in the skew diagram θ is a sequence of boxes (γ ₁ , . . . , γ N ) ∈ θ ^N such that for all 1 ≤ i ≤ N −1, γ i

and γ _i+1 have one common side. We say that θ is connected if given any two boxes γ, γ ^′ ∈ θ, there exists a path within θ connecting γ to γ ^′ . A ribbon is a connected skew diagram that contains no 2 × 2 block of boxes. Let ρ be a ribbon. The head (resp. tail) of ρ is the node γ = (i, j) ∈ ρ such that j − i is minimal (resp. maximal); we denote this node by hd(ρ) (resp.

tl(ρ)). If hd(ρ) = (i, j) and tl(ρ) = (i ^′ , j ^′ ), the height of ρ is the integer ht(ρ) := i − i ^′ ∈ N . Finally, the length of ρ is the number of boxes contained in ρ; we denote it by ℓ(ρ).

Example 3.2 On Figure 2 (see Section 3.4), the set of white squares represents the partition (4, 1) ; ρ, ρ ^′ and ρ ^′′ are three ribbons of respective heights 2, 1 and 0 and of respective lengths

4, 4 and 3. ⋄

A charged N -multi-partition is an element of Π ^N × Z ^N . If (λ, s) ∈ Π ^N × Z ^N is a charged multi-partition and s = (s ₁ , . . . , s N ), the content of the node γ = (i, j, b) ∈ λ is the integer

(23) cont(γ ) := s b + j − i.

If M ∈ N ^∗ , the residue modulo M of γ is

(24) res M (γ) := cont(γ) mod M ∈ Z /M Z . For all i ∈ Z , set

(25) N i (λ) := ♯{γ ∈ λ | res n (γ ) = i mod n} ;

this number depends on the multi-charge s. Define in a similar way N _i (θ) if θ is a skew diagram contained in a charged partition.

3.2 The bijection τ _l , the ordering ≺ and abaci

Throughout the proof of Theorem 2.11, we need a large amount of notation which we in- troduce here. In particular, we have to pass from l-multi-partitions (indexing the bases of the Fock space) to partitions (indexing the bases of the q-wedge space – see Section 4.1) and conversely. Following [U2], we achieve this using a bijection τ _l which can be described in a combinatorial way (see Definition 3.6). This map is a variant of the bijection associating to a partition its l-quotient and its l-core. We construct here τ _l using abaci; for another (equivalent) description of τ _l and examples, see [U2, Remark 4.2 (ii) and Example 4.3]. The bijection τ l is used in particular for defining the partial ordering ≺ on Π ^l _m mentioned in Section 2.2.2; see Definition 3.10.

3.2.1 Notation

The Euclidean algorithm shows that any integer k ∈ Z can be written in a unique way as

(26) k = c(k) + n(d(k) − 1) + nlm(k),

with c(k) ∈ [[1; n]], d(k) ∈ [[1; l]] and m(k) ∈ Z . Consider the map

(27) φ : Z → Z , k 7→ c(k) + nm(k).

φ enjoys the following obvious properties, which we need later: for all k, k ^′ ∈ Z , we have

φ(k) ≡ c(k) ≡ k (mod n), (28)

k < k ^′ , d(k) = d(k ^′ )

= ⇒ φ(k) < φ(k ^′ ), (29)

k ≤ k ^′ , φ(k) ≥ φ(k ^′ )

= ⇒ m(k) = m(k ^′ ).

(30)

For any r-tuple k = (k 1 , . . . , k r ) ∈ Z ^r , let

(31) c(k) := (c(k ₁ ), . . . , c(k r )) ∈ Z ^r ,

and define in a similar way d(k). The group S _r acts on the left on Z ^r by (32) σ.(k ₁ , . . . , k _r ) = (k _σ

(1) , . . . , k _σ

(r) ) ((k ₁ , . . . , k _r ) ∈ Z ^r , σ ∈ S _r ),

and a fundamental domain for this action is B := {(b ₁ , . . . , b r ) ∈ Z ^r | b ₁ ≥ · · · ≥ b r }. Let b(k) denote the element of B that is conjugated to d(k) under the action of S _r , W k be the stabilizer of b(k) (this is a parabolic subgroup of S _r ) and ω(k) be the element of maximal length in W k . Let W ^k be the set of minimal length representatives in the left cosets S _r /W k , and v(k) be the element in W ^k such that d(k) = v(k).b(k).

Example 3.3 Let n = 3, l = 2, r = 4 and k = (12,− 5, 2, 17). Then we have:

c(k) = (3, 1, 2, 2), d(k) = (2, 1, 1, 2), b(k) = (2, 2, 1, 1), v(k) = σ ₃ σ ₂ , ω(k) = σ ₁ σ ₃ . ⋄ Remark 3.4 Let k = (k ₁ , . . . , k r ) ∈ Z ^r . We can describe the action of v(k) ⁻¹ on k as follows.

Consider k as a word formed by the letters k i and for 1 ≤ d ≤ l, denote by w d the subword of k formed by the letters k i such that d(k i ) = d. Then we have v(k) ⁻¹ .k = w l · · · w ₁ . ⋄ 3.2.2 The bijection τ _l , the ordering ≺ and abaci

Definition 3.5 A 1-runner abacus is a subset A of Z such that −k ∈ A and k / ∈ A for all large enough k ∈ N . In a less formal way, each k ∈ A corresponds to the position of a bead on the horizontal abacus A which is full of beads on the left and empty on the right. Let A be the set of 1-runner abaci. If N ≥ 1, an N -runner abacus is an N -tuple of 1-runner abaci.

If A = (A ₁ , . . . , A N ) ∈ A ^N is an N -runner abacus, we identify A with the subset (33) {(k, d) | 1 ≤ d ≤ N, k ∈ A d } ⊂ Z × [[1; N ]].

⋄ To λ = (λ ⁽¹⁾ , . . . , λ ^(N) ) ∈ Π ^N and s = (s ₁ , . . . , s N ) ∈ Z ^N we associate the N -runner abacus

(34) A(λ, s) := {(λ ^(d) _i + s d + 1 − i, d) | i ≥ 1, 1 ≤ d ≤ N }.

One checks easily that the map

(35) (λ, s) ∈ Π ^N × Z ^N 7→ A(λ, s) ∈ A ^N is bijective.

Recall the definition of the maps k 7→ d(k) and k 7→ φ(k) from Section 3.2.1. Note that k ∈ Z 7→ φ(k), d(k)

∈ Z × [[1; l]] is a bijection.

Definition 3.6 The bijection τ l : Π × Z ∼ = A → Π ^l × Z ^l ∼ = A ^l is defined in terms of abaci by the formula

(36) τ l (A) :=

φ(k), d(k) k ∈ A ∈ A ^l (A ∈ A).

⋄ Remark 3.7 Let λ _l ∈ Π ^l , s l = (s 1 , . . . , s l ) ∈ Z ^l , λ ∈ Π and s ∈ Z satisfying the relation (λ l , s l ) = τ l (λ, s). Then we have s = s ₁ + · · · + s l . ⋄ Notation 3.8 Let s _l = (s ₁ , . . . , s _l ) ∈ Z ^l and s := s ₁ + · · · + s _l . Write

(37) λ l

s

←→ λ

if λ ∈ Π and λ l ∈ Π ^l _m are related by (λ l , s l ) = τ l (λ, s). We drop the s l in the notation if it is

clearly given by the context. ⋄

Example 3.9 Let n = 2, l = 3, m = 5 and s _l = (0, 0, −1). Then Figure 1 shows that (1, 1), (1, 1), (1) ^s

←→ (4, 3, 3, 2, 1).

-2 -3 -4 -6

-7 -8

-1

0 1

2 3

4 -5

-15

-5 2 8

-22 -16 -10 1

-28 -23

7 -11

-17 -4

-14 -16

4 9 10

-26 -21 -20 -15 -14 -9 -8 -2 3

-3

-14 -16

5 6 11 12

-24 -19 -18 -13 -12 -7 -6 0

-1

Figure 1: Computation of the bijection τ _l using abaci.

⋄

We now define a partial ordering ≺ on Π ^l _m as follows.

Definition 3.10 Let s l = (s 1 , . . . , s l ) ∈ Z ^l . Let λ _l , µ _l ∈ Π ^l _m and λ, µ ∈ Π be such that λ l

s

←→ λ and µ _l ←→ ^s

µ. We say that λ l precedes µ _l and write

(38) λ l µ _l

if µ dominates λ. In particular, by (21), λ and µ must be partitions of the same integer.

Note that the ordering depends on the multi-charge s l that we consider. Write λ l ≺ µ _l if

λ _l µ _l and λ _l 6= µ _l . ⋄

3.3 β-numbers and ribbons

Throughout this section we fix an integer s ∈ Z .

Definition 3.11 Let λ = (λ ₁ , λ ₂ , . . .) ∈ Π be a partition with at most r parts. The r-tuple (39) β _r (λ) := (λ ₁ + s, λ ₂ + s − 1, . . . , λ r + s − r + 1) ∈ Z ^r

is called the r−list of β-numbers associated to (λ, s) or (with a slight abuse of notation) the list or sequence of β -numbers associated to λ. The set of integers that form β _r (λ) is denoted

by B _r (λ). ⋄

With the notation of the definition above, note that β _r (λ) is a decreasing sequence of integers all greater than (or equal to) s + 1 − r. This sequence depends on the integer s we have fixed, but we do not mention it in our notation. Note that a partition λ is completely determined by its sequence of β -numbers. If r = |λ|, write more simply

(40) β(λ) := β _r (λ) and B(λ) := B _r (λ).

If f is a function defined on Z ^r , it is convenient to consider f as a function (still denoted by f ) defined on the set of partitions of r by the formula

(41) f (λ) := f (β(λ)) (λ ∈ Π, |λ| = r).

For example, we define this way for any partition λ the vectors c(λ), d(λ) and so on. See Section 3.2.1 for the corresponding notation.

In order to prove Theorem 2.11, we have to relate the adding/removal of a ribbon in a charged partition and the corresponding β-numbers. Let us recall a classical result on β-numbers (see e.g. [Mat1, Lemma 5.26]).

Lemma 3.12 Let ν and κ be two partitions with at most r parts, and let β _r (ν) = (α ₁ , . . . , α r ) and β _r (κ) = (β ₁ , . . . , β r ) denote the sequences of β -numbers associated to ν and κ respectively.

Then the following statements are equivalent.

(i) ν ⊂ κ, and ρ := κ/ν is a ribbon of length h.

(ii) There exist positive integers b and h such that B r (ν) = {β 1 , . . . , β _b−1 , β b −h, β b+1 , . . . , β r }.

In this case, b is the row number of the tail of ρ and h is the length of ρ. Let σ ∈ S _r denote the permutation obtained by arranging decreasingly the integers (β ₁ , . . . , β _b−1 , β b −h, β _b+1 , . . . , β r ).

Then we have ℓ(σ) = ht(ρ). Moreover, the content of the head of ρ is

(42) cont(hd(ρ)) = α _c = β _b − h,

where c is the row number of the head of ρ.

Proof. The proof of (i) ⇒ (ii) is easy. Conversely, assume that (ii) holds. Then we must have β b −h ≥ s + 1 − r, and there must exist b ≤ c ≤ r such that β c > β b − h > β _c+1 (if c = r, put β _c+1 := s − r). Note then that ν is obtained from κ by removing a ribbon ρ, where ρ ⊂ κ is the ribbon whose head is located at row c of κ and whose tail is located at row b of κ. ρ is actually a ribbon of length h. Moreover, with the notation of the statement of this lemma, we have σ.(β ₁ , . . . , β _b−1 , β b − h, β _b+1 , . . . , β r ) = (β ₁ , . . . , β _b−1 , β _b+1 , . . . , β c , β b − h, β _c+1 , . . . , β r ), hence σ is a cycle of length c − b = ht(ρ). Finally, the head of ρ has coordinates (c, ν _c + 1), so its content is equal to cont(hd(ρ)) = s + (ν c + 1) − c = α c = β b − h.

Example 3.13 Let s = 4, r = 5, κ = (6, 5, 3, 2, 2) and ν = (6, 2, 2, 2, 2). Then the skew diagram ρ := κ/ν is a ribbon and we have b = 2, c = 3 and h = 4. Moreover, we have β(κ) = (β ₁ , . . . , β ₅ ) = (10, 8, 5, 3, 2) and β(ν) = (β ₁ , β ₃ , β _b − h, β ₄ , β ₅ ) = (10, 5, 4, 3, 2). We have σ = (2, 3), hence ℓ(σ) = 1 = ht(ρ). The head of ρ has coordinates (3, 3), so its content

is cont(hd(ρ)) = 4 = β b − h. ⋄

Lemma 3.14 Let ν, κ ∈ Π be such that |ν| = |κ| = r and ν 6= κ. Let β(ν) = (α ₁ , . . . , α _r ) and β(κ) = (β ₁ , . . . , β r ) denote the sequences of β-numbers associated to ν and κ respectively.

Set ρ := ν/(ν ∩ κ) and ρ ^′ := κ/(ν ∩ κ).

1) Then, ρ and ρ ^′ are two ribbons if and only if ♯ B (ν) ∩ B(κ)

= r − 2. In this case, denote by

. h the common length of ρ and ρ ^′ , . y the row number of the tail of ρ ^′ , . y ^′ the row number of the head of ρ ^′ , . x ^′ the row number of the tail of ρ, and . x the row number of the head of ρ.

Then we have

{α i | i 6= x ^′ , y ^′ } = {β j | j 6= x, y}, (43)

cont(hd(ρ)) = β x = α x

− h and cont(hd(ρ ^′ )) = α y

= β y − h.

(44)

Let π ∈ S _r be the permutation obtained by arranging decreasingly the integers forming

B(ν ). Then we have ℓ(π) = ht(ρ) + ht(ρ ^′ ).

2) Assume that the conditions of 1) hold. Then we have the following equivalences, and moreover one of the two following cases occurs:

(i) y ≤ y ^′ < x ^′ ≤ x ⇐⇒ ν ⊳ κ, (ii) x ^′ ≤ x < y ≤ y ^′ ⇐⇒ κ ⊳ ν.

Proof. We prove 1) by applying the previous lemma to the pairs of partitions (ν ∩ κ, ν) and (ν ∩ κ, κ). Let us prove 2). The inequalities y ≤ y ^′ and x ^′ ≤ x are obvious. Since ρ ∩ ρ ^′ = ∅, one of the two following cases occurs: either y ^′ < x ^′ and then ν ⊳ κ, or x < y and then κ ⊳ ν.

This proves both implications ⇒, and since one of the two cases occurs, we get the desired

equivalences.

3.4 Definition of the matrix J ^<

Let R be a local ring, with unique maximal ideal ℘. We define in this section a matrix J = J _λ

,µ

λ

,µ

∈Π

, with coefficients in R, depending on parameters m, l ∈ N ^∗ and v, u ₁ , . . . , u _l ∈ R. This matrix is closely related to the matrix formed by the entries ν ℘ (g _λ

_,µ

) of [JM] (see (8)). Let λ _l = (λ ⁽¹⁾ , . . . , λ ^(l) ), µ _l = (µ ⁽¹⁾ , . . . , µ ^(l) ) ∈ Π ^l _m , and consider the following cases.

• Case (J ₁ ). Assume that λ l 6= µ _l and that there exist two integers d, d ^′ ∈ [[1; l]], d 6= d ^′ satisfying the following conditions: µ ^(d) ⊂ λ ^(d) , λ ^(d

⁾ ⊂ µ ^(d

⁾ , λ ^(b) = µ ^(b) for all integer b ∈ [[1; l]] \ {d, d ^′ }, and ρ := λ ^(d) /µ ^(d) and ρ ^′ := µ ^(d

⁾ /λ ^(d

⁾ are two ribbons of the same length b h. Let hd(ρ) = (i, j, d) denote the head of ρ and hd(ρ ^′ ) = (i ^′ , j ^′ , d ^′ ) denote the head of ρ ^′ . Set

(45) ε := (−1) ^ht(ρ)+ht(ρ

⁾ and

(46) J _λ

_,µ

:= u _d v ^j−i − u _d

v ^j

⁻ⁱ

^ε .

• Case (J ₂ ). Assume that λ _l 6= µ _l and that there exists d ∈ [[1; l]] such that λ ^(b) = µ ^(b) for all b 6= d, and ρ := λ ^(d) /(λ ^(d) ∩ µ ^(d) ) and ρ ^′ := µ ^(d) /(λ ^(d) ∩ µ ^(d) ) are two ribbons of the same length b h. By definition of ρ and ρ ^′ , we have ρ ∩ ρ ^′ = ∅, whence we get (depending on the relative positions of ρ and ρ ^′ ) that either λ ^(d) ⊳ µ ^(d) or µ ^(d) ⊳ λ ^(d) . Assume that λ ^(d) ⊳ µ ^(d) . Let ρ ^′′ ⊂ (λ ^(d) ∩ µ ^(d) ) be the ribbon obtained by connecting the tail of ρ to the head of ρ ^′ , excluding the two latter nodes (see Figure 2). Denote by hd(ρ) = (i, j, d) (resp. hd(ρ ^′ ) = (i ^′ , j ^′ , d ^′ ), resp. hd(ρ ^′′ ) = (i ^′′ , j ^′′ , d ^′′ )) the head of ρ (resp. ρ ^′ , resp. ρ ^′′ ), and finally set

(47) ε ₁ := (−1) ^ht(ρ)+ht(ρ

⁾ , ε ₂ := (−1) ^ht(ρ∪ρ

^)+ht(ρ

^∪ρ

⁾ and

(48) J _λ

_,µ

:= u _d (v ^j−i − v ^j

⁻ⁱ

) ^ε

. u _d (v ^j−i − v ^j

⁻ⁱ

) ^ε

.

If µ ^(d) ⊳ λ ^(d) , set J λ

,µ

:= J µ

,λ

.

• Case (J ₃ ). In all other cases, set

(49) J _λ

_,µ

:= 1.

00 00 11 11 00 00 11 11 00 00 11 11

00 00 11 11 00 00 11 11

00 00 11 11

00 00 11 11

00 00 11 11 00 00 11 11

00 00 11 11

00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 ^ρ

ρ ρ

’

’’

Figure 2: The ribbons ρ, ρ ^′ and ρ ^′′ (the nodes of (λ ^(d) ∩ µ ^(d) ) − ρ ^′′ are depicted in white).

We now define a matrix J = J ℘ = j λ

,µ

λ

,µ

∈Π

, with integer coefficients, by the formula

(50) j λ

,µ

:= ν ℘ (J λ

,µ

) (λ l , µ _l ∈ Π ^l _m ).

Now, let 6 be an arbitrary partial ordering on Π ^l _m and write λ _l < µ _l if λ _l 6 µ _l and λ _l 6= µ _l (λ l , µ _l ∈ Π ^l _m ). Recall the definition of the matrix J ^< = j _λ ^<

l

,µ

λ

,µ

∈Π

from (9) ; namely, put

(51) j _λ ^<

l

,µ

:=

( j λ

,µ

if λ _l < µ _l

0 otherwise (λ l , µ _l ∈ Π ^l _m ).

If we take 6 = E , then we get a matrix J ^⊳ whose entries correspond, up to conjugation

of multi-partitions, to the integers ν ℘ (g _λ

_,µ

) of [JM] (see (8)). Given a multi-charge s _l , we

shall also consider the matrix J ^≺ , where the ordering ≺ (depending on s l ) was introduced in

Definition 3.10. This is the matrix J ^≺ of Theorems 2.8 and 2.11. If s l is m-dominant (in the

sense of Definition 2.12), then the matrices J ^≺ and J ^⊳ coincide (see Proposition 5.12).

4 q-deformed higher-level Fock spaces

In this section we follow [U2], to which we refer the reader for more details. The vector spaces we consider here are over C (q), where q is an indeterminate over C .

4.1 q-wedge products and higher-level Fock spaces

Let s ∈ Z . Let Λ ^s denote the (semi-infinite) q-wedge space of charge s (this space is denoted by Λ ^s+

in [U2]). Λ ^s is an integrable representation of level l of the quantum algebra U q ( sl b _n ).

As a vector space, it has a natural basis formed by the so-called ordered q-wedge products.

These vectors can be written as

(52) u k = u k

∧ u k

∧ · · · ,

where k = (k _i ) _i≥1 is a decreasing sequence of integers such that k _i = s + 1 − i for i ≫ 0.

The basis formed by the ordered wedge products is called standard. More generally, we use the non-ordered wedge products; a non-ordered wedge product u k = u k

∧ u k

∧ · · · ∈ Λ ^s is indexed by a sequence of integers (k _i ) such that k _i = s + 1 − i for i ≫ 0, but we no longer require that (k i ) is decreasing. Any non-ordered wedge product can be written as a linear combination of ordered wedge products by using the so-called ordering rules, which are given in [U2, Proposition 3.16] and in a slightly different form in Proposition 4.4.

The vectors of the standard basis of Λ ^s can also be indexed by partitions as follows. Let u k = u _k

∧ u _k

∧ · · · ∈ Λ ^s be an ordered wedge product. For i ≥ 1 set λ i := k i − s + i − 1;

then λ := (λ 1 , λ 2 , . . .) is a partition. We then write u ^k = |λ, si. Note that if λ has at most r parts, then we have (k ₁ , . . . , k r ) = β _r (λ), which explains the definition of the β-numbers we gave in Definition 3.11.

Let F q [s l ] be the higher-level Fock space with multi-charge s l = (s ₁ , . . . , s l ) ∈ Z ^l [U2]. As a vector space, F q [s _l ] has a natural basis {|λ l , s _l i | λ _l ∈ Π ^l } indexed by l-multi-partitions.

If s = s ₁ + · · · + s _l , then F _q [s _l ] can be identified with a subspace of Λ ^s by the embedding F q [s l ] ֒ → Λ ^s , |λ l , s l i 7→ |λ, si, where λ is the partition such that λ l ↔ λ (see Notation 3.8 for the meaning of ↔). We make from now on this identification; in fact, Λ ^s is isomorphic to the direct sum of all the F _q [t _l ]’s, where t _l is any l-tuple of integers summing to s. Thus, the vectors of the standard basis of Λ ^s can also be indexed by charged l-multi-partitions.

4.2 The involution

In order to define the canonical basis of Λ ^s , we equip this space with an involution .

Definition 4.1 The involution of Λ ^s is the C -vector space automorphism that maps q to

q ⁻¹ and that acts on the standard basis of Λ ^s as follows [U2, Proposition 3.23 and Remark

3.24]. Let λ ∈ Π be a partition of r, and k = (k i ) ∈ Z ^N

be such that u ^k = |λ, si. Then

(53) |λ, si := (−1) ^{κ( d (λ))} q ^{κ( d (λ))−κ( c (λ))} (u _k

∧ · · · ∧ u _k

) ∧ u _k

∧ u _k

∧ · · · ,

where for any a = (a 1 , . . . , a r ) ∈ Z ^r , κ(a) is the integer defined by (54) κ(a) := ♯{(i, j) ∈ N ² | 1 ≤ i < j ≤ r, a i = a j },

and c(λ) and d(λ) are defined in Section 3.2.1. ⋄ We can straighten the non-ordered wedge product in the right-hand side of (53) in order to express it as a linear combination of ordered wedge products.

One checks that preserves the subspace

(55) F _q [s _l ] _m := M

λ

∈Π

C (q) |λ l , s _l i ⊂ F _q [s _l ].

Definition 4.2 Define a matrix A(q) = a _λ

_,µ

(q)

λ

,µ

∈Π

with entries in C (q) by

(56) |µ _l , s _l i = X

λ

∈Π

a _λ

_,µ

(q) |λ l , s _l i (µ _l ∈ Π ^l _m ).

⋄ The matrix A(q) depends on n, l, s _l and m. The ordering rules show that A(q) is unitriangular with respect to , that is

(57) a λ

,µ

(q) 6= 0 ⇒ λ l µ _l and a λ

,λ

(q) = 1 (λ l , µ _l ∈ Π ^l _m ).

The same rules also imply that A(1) is the identity matrix.

4.3 Uglov’s canonical basis

Since the matrix A(q) of the involution of F q [s l ] m is unitriangular, a classical argument can be used to prove the following result.

Theorem 4.3 ([U2]) There exists a unique basis {G(λ l , s _l ) | λ _l ∈ Π ^l _m } of F _q [ s _l ] _m satisfying both following conditions:

(i) G(λ l , s _l ) = G(λ l , s _l ), (ii) G(λ l , s _l ) − |λ l , s _l i ∈ M

µ

∈Π

q C [q] |µ _l , s _l i.

Definition 4.4 The basis {G(λ l , s l ) | λ _l ∈ Π ^l _m } is called the canonical basis of F q [s l ] m . Define a matrix ∆(q) = ∆ _λ

,µ

(q)

λ

,µ

∈Π

with entries in C [q] by

(58) G(µ _l , s _l ) = X

λ

∈Π

∆ _λ

_,µ

(q) |λ l , s _l i (µ _l ∈ Π ^l _m ).

⋄

The matrix ∆(q) depends on n, l, s l and m. By Condition (ii) of Theorem 4.3, the matrix

∆(q) is also unitriangular with respect to . By [U2, Theorem 3.26], the entries of ∆(q) can be expressed as Kazhdan-Lusztig polynomials related to parabolic modules of an affine Hecke algebra of type ˜ A, so by [KT], these entries are in N [q].

4.4 Another basis of Λ ^s . Ordering rules.

The ordering rules (R ₁ )-(R ₄ ) from [U2, Proposition 3.16] do not give at q = 1 anticommuting relations like u _k

∧ u _k

= −u k

∧ u _k

, because of the signs involved in Rules (R ₃ ) and (R ₄ ).

To fix this, we introduce another basis of Λ ^s that differs from the standard basis only by signs. The basis we consider here is actually the basis of ordered wedge products introduced in [U1]. Λ ^s is graded by

(59) deg(|λ, si) := |λ| (λ ∈ Π).

Let u k = u _k

∧ u _k

∧ · · · ∈ Λ ^s be a (not necessarily ordered) wedge product of degree r. Set

(60) v k = v _k

∧ v _k

∧ · · · := (−1) ^ℓ(v(k

^,...,k

⁾⁾ u k

and similarly v k

∧ · · · ∧ v k

:= (−1) ^ℓ(v(k

^,...,k

⁾⁾ u k

∧ · · · ∧ u k

,

where v(k ₁ , . . . , k r ) ∈ S _r is defined in Section 3.2.1. (If k = (k ₁ , . . . , k r ) ∈ Z ^r , we hope that the reader will make easily the difference between the permutation v(k) ∈ S _r and the wedge product v k = v _k

∧ · · · ∧ v _k

.) We say that the wedge product v k is ordered if so is u k . It is straightforward to see, using the ordering rules for the u ^k ’s given by [U2, Proposition 3.16], that the ordering rules for the v k ’s are given by the following proposition.

Proposition 4.5

(i) Let k ₁ ≤ k ₂ , and γ ∈ [[0; nl − 1]] (resp. δ ∈ [[0; nl − 1]]) denote the residue of c(k ₂ ) − c(k ₁ ) resp. of n(d(k ₂ ) − d(k ₁ ))

modulo nl. Then we have

(R ₁ ) v _k

∧ v _k

= −v k

∧ v _k

if γ = δ = 0,

(R ₂ )

v _k

∧ v _k

= −q ⁻¹ v _k

∧ v _k

−(q ⁻² − 1) X

i≥1

q ⁻²ⁱ⁺¹ v k

−nli ∧ v k

+nli

+(q ⁻² − 1) X

i≥0

q ⁻²ⁱ v _k

_−γ−nli ∧ v _k

_+γ+nli

if γ > 0, δ = 0,

(R ₃ )

v _k

∧ v _k

= −qv k

∧ v _k

−(q ² − 1) X

i≥1

q ²ⁱ⁻¹ v k

−nli ∧ v k

+nli

+(q ² − 1) X

i≥0

q ²ⁱ v _k

_−δ−nli ∧ v _k

_+δ+nli

if γ = 0, δ > 0,

(R ₄ )

v k

∧ v k

= −v k

∧ v k

−(q − q ⁻¹ ) X

i≥1

q ²ⁱ − q ⁻²ⁱ

q + q ⁻¹ v k

−nli ∧ v k

+nli

−(q − q ⁻¹ ) X

i≥0

q ²ⁱ⁺¹ + q ⁻²ⁱ⁻¹

q + q ⁻¹ v _k

_−γ−nli ∧ v _k

_+γ+nli +(q − q ⁻¹ ) X

i≥0

q ²ⁱ⁺¹ + q ⁻²ⁱ⁻¹

q + q ⁻¹ v _k

_−δ−nli ∧ v _k

_+δ+nli +(q − q ⁻¹ ) X

i≥0

q ²ⁱ⁺² − q ⁻²ⁱ⁻²

q + q ⁻¹ v k

−γ−δ−nli ∧ v k

+γ+δ+nli

if γ > 0, δ > 0,

where the sums range over the indices i such that the corresponding wedge products are or- dered.

(ii) The rules from (i) are valid for any pair of adjacent factors of the q-wedge product

v k = v _k

∧ v _k

· · · .

Let us end this section by a useful piece of notation.

Notation 4.6 Let ν ∈ Π be a partition of r and σ ∈ S _r . Set

(61) u _σ.ν := u _σ.β(ν) and similarly v _σ.ν := v _σ.β(ν)

(these are wedge products of r factors each). We say that u _σ.ν (resp. v _σ.ν ) is obtained from

u ν ( resp. v ν ) by permutation. ⋄

PART C: Proof of Theorem 2.11

We now start the proof of Theorem 2.11. In Section 5, we give a simpler expression for

the entries of the matrix J ^≺ (see Proposition 5.8). In Section 6, we compute the derivative

at q = 1 of the involution of F q [s l ] m in terms of good sequences that we introduce in

Definition 6.4; the result is given in Proposition 6.8. We compare both expressions in Section

7 in order to complete the proof. Apart from this, we compare in Section 5.2 the matrices

J ^≺ and J ^⊳ when the multi-charge s _l is m-dominant.

Notation for Part C From now on, we consider the modular system (R, K, F ) (together with the prime ideal ℘) with parameters defined in Section 2.2.1. These parameters depend on n, l, m and s _l = (s ₁ , . . . , s _l ) ∈ L(r 1 , . . . , r _l ) that we have fixed. Recall that to s _l we associated a partial ordering ≺ (see Definition 3.10) and a relation ↔ (see Notation 3.8).

Finally, put s := s ₁ + · · · + s l . ⋄

5 Expression of the matrices J ^≺ and J ^⊳

5.1 The matrices J and J ^≺

We first give, with our choice of parameters, a simpler expression for J . Lemma 5.1

1) Assume that (λ l , µ _l ) satisfies the conditions (J ₁ ). Then we have (62) j λ

,µ

=

(−1) ^ht(ρ)+ht(ρ

⁾ if res n (hd(ρ)) = res n (hd(ρ ^′ )),

0 otherwise.

2) Assume that (λ l , µ _l ) satisfies the conditions (J 2 ). Then we have (63) j _λ

_,µ

= (−1) ^ht(ρ)+ht(ρ

⁾ ε,

where

(64) ε :=

 



1 if res _n (hd(ρ)) = res _n (hd(ρ ^′ )) and b h 6≡ 0 (mod n),

−1 if res _n (hd(ρ)) 6= res n (hd(ρ ^′ )) and b h ≡ 0 (mod n), 0 otherwise,

and b h is the common length of ρ and ρ ^′ .

Proof. Let us prove 1). With our choice of parameters, we have j λ

,µ

= (−1) ^ht(ρ)+ht(ρ

⁾ ν ℘ (P λ

,µ

(x)) with P λ

,µ

(x) := u d x ^l(j−i) − u d

x ^l(j

⁻ⁱ

⁾ . Note that

P _λ

_,µ

(x) = ξ ^a

x ^a

− ξ ^a

x ^a

,

where ξ ∈ C is a primitive nl-th root of unity and a ₁ := dn, a ₂ := ls _d − dn + l(j − i), a ₃ := d ^′ n and a ₄ := ls _d

− d ^′ n + l(j ^′ − i ^′ ). Using the fact that ν _℘ (x ^N ) = 0 for all N ∈ Z and x ^N P λ

,µ

(x) ∈ C [x] for a suitable N ∈ Z , we get

ν _℘ (P _λ

_,µ

(x)) ≥ 0,

ν _℘ (P _λ

_,µ

(x)) ≥ 1 ⇐⇒ P _λ

_,µ

(ξ) = 0, ν ℘ (P λ

,µ

(x)) ≥ 2 ⇐⇒ P λ

,µ

(ξ) = P _λ ^′

l

,µ

(ξ) = 0.